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The cost function of eqn (518) can be motivated in two contexts First, in a deterministic sense, it is desirable to minimize some norm of the error between the measurements Y and the estimated measurements = H The cost function is a weighted two norm In a general approach, Y x some measurements may be known to be more accurate than others The measurement accuracy is characterized by R 1 Therefore, it is natural to select (519) W = R 1 to give the least weighting to the most uncertain measurements Second, in a probabilistic sense, given the assumption that the measurement noise is Normally distributed, the probability density for x given the measurements Y is 1 Y Hx p(x : Y) = A exp 2 R 1 Y Hx (520)

a 1 = 1/(a1) = 1/a a 3 = 1/(a 3) a 20 = 1/(a 20) As before, a can be almost any expression you can imagine Note that the 1st power of any quantity is the same thing as its reciprocal (multiplicative inverse) You ll often see this notation used because it can be a lot less messy than writing 1, then a slash, and then a complicated expression Here are some examples of numbers or quantities raised to negative integer powers: 4 2 x 4 where x 0 7 (k + 4) where k 4 4 (abc) where a 0, b 0, and c 0 (m /n) 12 where m 0 and n 0 2 (x 2x + 1) 5 where x 1

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The value of x that maximizes p(x : Y) is the maximum likelihood estimate (MLE) of x and coincides with the minimum of JW LS when W = R 1 For m < n, the problem will be under-determined and an in nite number of solutions will exist If m > n then the problem may be overdetermined, in which case no exact solution will exist In the latter case, minimization of the objective function in eqn (518) will result in the es timate x that has the minimum weighted 2-norm of y = [ y1 , , ym ] where yi = yi Hi x The following text will build up to the Kalman Filter by discussing sequentially the following set of questions: 1 What is the formula for the WLS estimate of x (ie, the minimum of eqn (518)) 2 Given that the WLS estimate xm has been calculated for a set of m measurements, if an additional measurement ym+1 becomes available, can xm be adjusted to e ciently produce the new WLS estimate xm+1 3 If x Rn is the state of a dynamic system, which is free to change according to xk = k xk 1 + Gk uk 1 + k 1 where k Rn n is known and k Rn is a zero mean random vector, how can the estimate of x at time k 1 and the measurement yk be combined to provide an optimal estimate of x at time k As the above discussion shows, each question is a natural extension of the one before it and as the following sections show, the solutions to these problems are intimately related The Kalman Filter is the solution to the last problem

The constraints are imposed, as with the 0th power, to keep the powerized quantities from being equal to 0 If you take a negative integer power of 0, you end up with 1/0, and that s not defined For example, 0 7 = 1/(07) = 1/0

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